## Ergodicity and stability of non-stationary queueing system.(Russian, English)Zbl 1050.60075

Teor. Jmovirn. Mat. Stat. 68, 1-10 (2003); translation in Theory Probab. Math. Stat. 68, 1-10 (2004).
The authors deal with the non-homogeneous Markov process (birth and death process) $$X(t)$$, $$t\geq0$$, with the intensity matrix $$A(t)=\{a_{ij}(t)\}$$, $$t\geq0$$,
$a_{ij}(t)= \begin{cases} \lambda_{i-1}(t),&\text{if }j=i-1,\\ \mu_{i+1}(t),&\text{if }j=i+1,\\ -(\lambda_{i}(t)+\mu_{i}(t)),&\text{if }j=i,\\ 0,&\text{if }\| i-j\|>1, \end{cases}$
where $$\lambda_{n}(t)$$, $$n=0,1,\ldots,N$$, is the intensity of birth; $$\mu_{n}(t)$$, $$t\geq0$$, $$n=0,1,\ldots,N$$, is the intensity of death. The authors consider the case $$\lambda_{n}(t)=\lambda_{n}a(t)$$, $$\mu_{n}(t)=\mu_{n}b(t)$$, $$t\geq0,\;n=0,1,\ldots,N$$. The problems of ergodicity and stability for $$X(t)$$ are investigated. As examples queueing systems close to $$M_{t}/M_{t}/S$$ and $$M_{t}/M_{t}/S/0$$ are considered.

### MSC:

 60J27 Continuous-time Markov processes on discrete state spaces 60J80 Branching processes (Galton-Watson, birth-and-death, etc.) 60K25 Queueing theory (aspects of probability theory)